L(s) = 1 | + (2.13 − 0.656i)5-s + 2.27·7-s + 3·9-s − 2.15i·11-s + 8.24i·17-s − 4.35i·19-s + 4·23-s + (4.13 − 2.80i)25-s + (4.86 − 1.49i)35-s − 10.8·43-s + (6.41 − 1.97i)45-s − 10.2·47-s − 1.82·49-s + (−1.41 − 4.59i)55-s + 3.72·61-s + ⋯ |
L(s) = 1 | + (0.955 − 0.293i)5-s + 0.859·7-s + 9-s − 0.648i·11-s + 1.99i·17-s − 0.999i·19-s + 0.834·23-s + (0.827 − 0.561i)25-s + (0.821 − 0.252i)35-s − 1.65·43-s + (0.955 − 0.293i)45-s − 1.49·47-s − 0.260·49-s + (−0.190 − 0.619i)55-s + 0.476·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.466692999\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.466692999\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.13 + 0.656i)T \) |
| 19 | \( 1 + 4.35iT \) |
good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 2.27T + 7T^{2} \) |
| 11 | \( 1 + 2.15iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 8.24iT - 17T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 3.72T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2.98iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.438324082223123407704906581942, −8.585983041219050278760316857460, −8.050967702443631602490645916224, −6.86484106580748464882766542627, −6.23142674310024969192902607324, −5.20088839975652228743514485666, −4.57965804675782730634782857793, −3.42722780515022407641659120709, −2.00952701995217277751861758974, −1.24498120177484434051810935034,
1.32846950626729400691445266186, 2.19458815306264168957882313035, 3.39310593923467888935110964938, 4.86976979669395095926094719071, 5.01967165038520020020600042291, 6.36796656931150085371164308062, 7.09999100771115145389848631737, 7.73178287015560655242835827751, 8.815510856741539104385295943950, 9.767765955925962179640975394246